Let $z$ and $w$ be complex numbers such that $|z + 1 + 3i| = 1$ and $|w - 7 - 8i| = 3.$  Find the smallest possible value of $|z - w|.$
Answer: Let $a = -1 - 3i$ and $b = 7 + 8i.$  Then $z$ lies on the circle centered at $a$ with radius 1, and $w$ lies on the circle centered at $b$ with radius 3.

[asy]
unitsize (0.4 cm);

pair A, B, Z, W;

A = (-1,-3);
B = (7,8);
Z = A + dir(110);
W = B + 3*dir(210);

draw(A--B);
draw(Circle(A,1));
draw(Circle(B,3));
draw(A--Z--W--B);

dot("$a$", A, SW);
dot("$b$", B, NE);
dot("$z$", Z, NW);
dot("$w$", W, dir(180));
[/asy]

By the Triangle Inequality,
\[|a - z| + |z - w| + |w - b| \ge |a - b|,\]so
\[|z - w| \ge |a - b| - |a - z| - |w - b|.\]We have that $|a - b| = |(-1 - 3i) - (7 + 8i) = |-8 - 11i| = \sqrt{185}.$  Also, $|a - z| = 1$ and $|w - b| = 3,$ so
\[|z - w| \ge \sqrt{185} - 4.\]Equality occurs when $z$ and $w$ are the intersections of the circles with the line segments connecting $a$ and $b.$

[asy]
unitsize (0.4 cm);

pair A, B, Z, W;

A = (-1,-3);
B = (7,8);
Z = intersectionpoint(Circle(A,1),A--B);
W = intersectionpoint(Circle(B,3),A--B);

draw(A--B);
draw(Circle(A,1));
draw(Circle(B,3));

dot("$a$", A, SW);
dot("$b$", B, NE);
dot("$z$", Z, E);
dot("$w$", W, S);
[/asy]

Hence, the smallest possible value of $|z - w|$ is $\boxed{\sqrt{185} - 4}.$